Publisher Summary This chapter presents a study on the Uricase system and pattern formation in enzyme systems. The pattern formation means the stable spatially nonuniform concentration profiles supersede spatially uniform concentration profiles that become unstable as a bifurcation parameter crosses a critical value. The systems under consideration are open systems where an external reservoir supplies a flow of substrates, the substrates diffuse with distinct diffusion coefficients and they are consumed in an enzyme reaction. In the absence of diffusion constraints in the reactive medium, an equilibrium state settles with the same concentration values in all the cells. This so called “trivial steady state” is stable, and remains stable as long as relatively small diffusion constraints are imposed to the substrates within the reactive medium. This uniform distribution of diffusible reacting substrates loses stability as the diffusion constraints cross a critical value, and a stable patterned distribution of the substrates emerge. This is the so-called “Turing effect,” also known as “diffusion-induced instability” or “diffusive instability”. The chapter presents a single cell or compartment comprising a homogeneous solution of uricase, uric acid, and oxygen. Uricase is an enzyme that catalyzes the consumption of uric acid and oxygen according to the stoichiometric scheme.